On Wrapping Spheres and Cubes with Rectangular Paper - Discrete and Computational Geometry and Graphs - page 38

2 , a rectangle with x =2 √ 2 / √ n 2 +4 Lower Boun d 6. For any integer n ≥

wraps a 1 / (2 √ 2) -cube.

The proof of this proposition is almost identical to that of Lower Bound 5 .
Figure 4 should give the reader the core ideas. The principal difference is that
the bottom edge no longer maps to a single point, so the dissection is more
constrained. Combining bounds on the middle and bottom wrappings in Fig. 4
yields our lower bound. Interestingly, Lower Bound 6 for n = 2 reproduces the
square folding by Catalano-Johnson and Loeb [ 4 ].

4.3 Strip Folding

Strip folding is a technique introduced in [ 3 ] that weaves a narrow strip of paper
back and forth to cover a surface. This section sketches new strategies for strip
folding that produce superior bounds on the sphere and the cube.

Cubes. Refer to Fig. 5 . Here we present a new technique for strip folding on
the cube that is more ecient than that presented in [ 3 ]. The general strategy
consists of 3 parts resembling an algorithm more than a function:

Fig. 5. Strip wrapping a cube. 3D diagram (left) and edge unfolding (right).

- Spiral around the 4 vertical faces of the cube (sides).
- Fold the excess over onto top and bottom faces.
- Try two different methods of doubling back and forth using turn gadgets (as
seen in [ 3 ]) to cover the rest of the top and bottom faces.

We parameterize in terms of n , the number of times the top of the strip
switches faces while covering the sides. For ease of computation, we require
integral n .
The excess folded onto the top and bottom leaves a w

× h rectangle uncovered

( n− 5) S
√ 1+ n 2

( n− 3) S
√ 1+ n 2

on each, where w = and h = . In Fig. 5 , the bold lines indicate

these w

× h rectangles.